Spatial Control of 2D Nanomaterial Electronic Properties Using Chiral Light Beams

Single-layer two-dimensional (2D) nanomaterials exhibit physical and chemical properties which can be dynamically modulated through out-of-plane deformations. Existing methods rely on intricate micromechanical manipulations (e.g., poking, bending, rumpling), hindering their widespread technological implementation. We address this challenge by proposing an all-optical approach that decouples strain engineering from micromechanical complexities. This method leverages the forces generated by chiral light beams carrying orbital angular momentum (OAM). The inherent sense of twist of these beams enables the exertion of controlled torques on 2D monolayer materials, inducing tailored strain. This approach offers a contactless and dynamically tunable alternative to existing methods. As a proof-of-concept, we demonstrate control over the conductivity of graphene transistors using chiral light beams, showcasing the potential of this approach for manipulating properties in future electronic devices. This optical control mechanism holds promise in enabling the reconfiguration of devices through optically patterned strain. It also allows broader utilization of strain engineering in 2D nanomaterials for advanced functionalities in next-generation optoelectronic devices and sensors.


Set up for I-V measurements of monolayer graphene
An image of the Graphenea card where the GFET (graphene field effect transistor) is incorporated into is shown in Figure S1a.This includes switches for the drain and the source, and individual sources for each FET device.Each graphene channel can be selected with the external individual source switches.A schematic of the electric circuit is shown in Figure S1b.A schematic of the optical set up for laser illumination of the monolayer graphene is displayed in Figure S2 (see Methods for full description).

Beam profile of 405nm laser
Measured beam profiles for  = 0 and  = 2 beams are depicted in Figure S3a.The profiles were fitted to the intensity function () corresponding to the electric field of the LG beam squared.Recall that () =  0 at the focal point  = 0, where  0 is the beam waist, so the intensity function becomes: is the peak height,  is the beam radius and  0 is the profile centre. is the topological charge, so for  = 0 the intensity function reduces to that of a Gaussian beam.The fitted data and extracted beam waists and diameters for  = 0 and  = 2 beams are given in FigureS3b and c, respectively.For an LG beam with  ≠ 0 the beam diameter may be defined as the separation between intensity maxima, 1 though a more convenient measure may be the root mean squared intensity, as this definition may be applied also to the  = 0 Gaussian beam. 2 For an LG beam with radial index  = 0 and azimuthal index , the latter definition gives a diameter of  =

AFM of graphene when illuminated with a Gaussian beam
Supplementary AFM measurements collected with a Gaussian beam ( = 0) are shown in Figure S6.
Figure S6 shows that the changes in contrast after illuminating with Gaussian beam are much smaller than those observed with LG beam ( = 2) in Figure 3 of the main manuscript.Furthermore, the rootmean-square roughness (R RMS ) only increases by 0.3 nm.

Finite-element electromagnetic simulations of Laguerre-Gaussian beams in air
Electromagnetic simulations of Laguerre-Gaussian beams (see Methods for details) were first tested in air. Figure S7 shows that the size of the LG beam increases for larger topological charge as expected, and the rotational sense of the phase fronts is reversed upon changing the sign of the topological charge.The validity of the simulations was also checked by simulating the angular momentum (AM) flux 3 in air.The results from these simulations show that the AM flux increases linearly as a function of topological charge (Figure S8a) and power (Figure S8b) as expected.

Finite-element simulated laser heating of Laguerre-Gaussian beams in monolayer graphene on SiO 2 /Si
Laser heating simulations (see Methods for details) were performed for monolayer graphene on SiO 2 /Si, and the results are shown in Figure S9.It is obtained that LG beams for increasing topological charges produce smaller temperature rises than a Gaussian beam, as the power is distributed over a larger area.

Photoluminescence spectroscopy set up
A schematic of the photoluminescence (PL) set up used to conduct PL spectroscopy of monolayer WS 2 is shown in Figure S10 (see Methods for full description).

Photoluminescence spectroscopy of monolayer WS 2
The changes observed for exfoliated monolayer with LG beams are displayed in Figure S11.

Response of monolayer WS 2 to temperature
Heating of monolayer WS 2 was carried out using a commercial heating table (Linkam Scientific CO 102).Heating was performed from 20 °C to 50 °C in increasing steps of 10 °C.Before collecting the PL spectrum, the sample was left to stabilise at that temperature for 1h.
The changes in the PL spectra associated with heating the CVD-grown monolayer WS 2 on Al 2 O 3 are shown in Figure S15a, showing a red shift of the PL emission.The energies of the neutral exciton and trion decrease as a function of temperature at approximately the same rate (Figure S15b), contrary to our observations with OAM (Figure 4g,h in the main manuscript).Furthermore, the trion-toneutral exciton intensity ratio only changes significantly for a temperature of 40 ℃ and above (Figure S15c).Experimental heating of an exfoliated monolayer flake is shown in Figure S16, also showing that the energy of the neutral exciton and trion decrease at the same rate within fitting error, and the trionto-neutral exciton ratio only changes significantly for a temperature of 40 ℃ and above.Laser heating simulations (see Methods for details) were performed for monolayer WS 2 on the two different substrates used, Al 2 O 3 and SiO 2 /Si, and the results are shown in Figure S17 and Figure S18, respectively.In both cases, it is obtained that LG beams for increasing topological charges produce smaller temperature rises than a Gaussian beam, as the power is distributed over a larger area.

Power-dependent photoluminescence on CVD-grown and exfoliated monolayer WS 2
The power-dependent PL spectrum of CVD-grown full-coverage monolayer WS 2 on Al 2 O 3 is shown in Figure S19a, where the energy and trion-to-neutral exciton intensity ratio do not change with laser power, in agreement with the literature. 4In exfoliated monolayer WS 2 , the trion is particularly dominant at large excitation powers (Figure S19b), in agreement with the literature. 5As LG beams have less power density than Gaussian beams (Supplementary section 9), the red shift and appearance of a shoulder at lower energy with LG beam illumination cannot be accounted for by a change in power density of the beam in the exfoliated monolayer, as reduced power density would result in a blue shift, not a red shift.

Derivation of momentum flux density inside a medium
In this section, we use the local continuity of momentum within a dielectric medium to derive an expression for the electromagnetic momentum flux tensor inside the medium.The conservation of the -th component of electromagnetic momentum inside the medium can be expressed using the continuity equation: where   is the momentum density,   is the Lorentz force density, and   is the momentum flux tensor (the -th component of   is the flux of the  component of momentum in the  direction).A sum over repeated indices is implied.If we now write down expressions for the momentum density , and the Lorentz force density, , we can obtain an expression for   from the requirement that equation ( 1 ) is satisfied.
In free space, the momentum density of light is straightforwardly given by  = 1  2  × . 6However, inside the medium the definition of electromagnetic momentum is more subtle, and different formulations exist. 7Here we use the Abraham form, and take the momentum density of light inside the medium to be  = 1 The Lorentz force density can be written where  is the polarisation density * . 8bstituting in these expressions for momentum density and Lorentz force density, equation ( 1 ) becomes: where a dot above a symbol indicates a time derivative.Assuming that the material is nonmagnetic, we have  = 1  0 , so the left-hand side of equation ( 3 ) can be rewritten as: We then use the definition of the displacement field,  =  0  + , to remove the polarisation field from the expression above: We now use Maxwell's equations ∇ ×  = - and ∇ ×  =  = 1  0 ∇ ×  (where we assume that the medium is non-magnetic): Finally, we use the identity     =     -    to write: Our task is now to write the expression ( 7 ) as the divergence of a second rank tensor.It is straightforward to show that this is accomplished if we define since -∇    is: Assuming there are no free charges, we can use Maxwell's equations ∇ •  = 0 and ∇ •  = 0, so the equality of equations ( 7)( 7) and ( 9)( 9 ) immediately follows.

Derivation of angular momentum continuity equation inside a medium
Angular momentum  is the cross product of the position vector  with the linear momentum : Similarly, we can define the angular momentum flux density as the "cross product" of the position vector with the momentum flux tensor. 3The -component of the angular momentum flux in the  direction is: Using the expression derived for the linear momentum flux density in supplementary section 11 (equation ( 8)), the angular momentum flux density in the medium is given by To obtain a continuity equation for the angular momentum of light, we begin by taking the cross product of the position vector  and the linear momentum continuity equation ( The first term in equation ( 13 ) represents the time derivative of the angular momentum density The second two terms are  × , where  is the Lorentz force density given in equation ( 2 ), and so these correspond to a torque density.However, this is not the total torque density.The force density was defined as the force acting on the centre of each electric dipole, and therefore the torque density derived from this does not include the torque on each individual dipole about its own centre.We might expect an extra torque, with a form like   =  × , which would act to orient each dipole to align with the electric field.
Finally, we note that the term on the right-hand side is not quite equal to the divergence of the angular momentum flux density.The divergence of  is equal to the divergence of the cross product of  and , but so far the right-hand side of equation ( 13 ) is the cross product of  with the divergence of .We can see that the divergence of  will have an extra contribution, due to the product rule when the divergence is taken: The Because     = 0, the part involving  2 and  2 vanishes, and because   is antisymmetric, the contraction with the symmetric tensor     also vanishes.can then use  =  0  +  to write: where we have again used the antisymmetry of   to write       = 0. Equation ( 14 ) therefore becomes: We can therefore see that if we add ( × )  to both sides of equation ( 13 ), we obtain the continuity equation:

Calculation of net forces and torques imparted on a material
Recall that the continuity equation for linear momentum of light is given by (Supplementary Section 11): where   is the momentum flux density given in equation ( 8 ).If an experiment is performed with a continuous source of monochromatic light, then it is possible to time-average the continuity equation over a cycle.Under this averaging, the term corresponding to the time derivative of the Poynting vector vanishes, and we are therefore able to equate the divergence of the momentum flux density with the force on the dielectric, This can be rewritten with the divergence theorem to find the force on a volume in terms of the integral of the momentum flux through the surface enclosing the volume, where   is a unit vector in the direction of the surface element .Therefore, the force imparted by the beam at an interface can be calculated by considering the surface integral of   over a surface enclosing the interface; this could, for example, be the difference in momentum fluxes through a plane just before the interface and one just after.
Similarly, a net torque can be obtained from the surface integral of the angular momentum flux density in the same way as a force can be obtained from the linear momentum.If a beam is propagating in the  direction normal to an interface, then the torque about this propagation direction will be given by the difference in the fluxes of the  component of angular momentum through two planes on either side of the interface.The relevant component of the angular momentum flux density tensor, to be evaluated on either side of the interface, is: where we have made use of the fact that   = 0.This quantity is in essence the cross product of position  and the momentum flux density.
We now compute the cycle-average of equation ( 22 ) for monochromatic fields.We begin by expressing the real electric and magnetic field components   and   in terms of complex fields ℰ  and ℬ  : =  ℬ  =    exp( -) , with complex amplitudes   and   .Maxwell's equations in the medium in the absence of sources are: Note  =  and  = .
In terms of the complex fields, Maxwell's equations become ∇ ×  - = 0, ( 31 ) From the last two equations we obtain: The cycle-averaged angular momentum flux density is given by: For a linearly-polarised b eam (i.e.no spin angular momentum), the AM flux equals to the orbital angular momentum (OAM) flux.

Figure
Figure S1.(a) Image of the card (Graphenea) where the GFET can be incorporated.(b) Electrical circuit of the GFET.

Figure S2 .
Figure S2.Schematic of the optical set up used to illuminate the GFET with Gaussian and LG beams and conduct I-V curves.The LG beam is generated by inserting a spiral phase plate ( = , 405 nm, Vortex Photonics) along the optical path.The beam is directed towards the sample with a 50:50 beam splitter and a 10X objective (NA=0.25).

Figure
Figure S3.(a) Measured beam profiles for  =  (black) and  =  beams (red) used to conduct conductance measurements, Raman spectroscopy and AFM of monolayer graphene.(b) Experimental Gaussian beam profile (black) fitted to the intensity expression with  =  (blue).(c) Experimental LG beam profile (black) fitted to the intensity expression with  =  (blue).For (b) and (c), the extracted beam waist   and beam diameter  are indicated underneath.

Figure S4 .
Figure S4.Raman spectra collected at the illuminated area with an LG beam ( = ) after a relaxation period >48 hours (black) and outside illuminated area (green).The spectra have been normalised to the maximum intensity of the 2D-band.

Figure S5 .
Figure S5.(a,b) Raman mapping of the G-band position before illumination (a) and after Gaussian beam ( = ) illumination (b) (scale bar: m).The approximate illuminated area in panel (b) is indicated with a white dotted circle.(c) Raman spectra for illuminated (black) and unilluminated (green) areas collected at the positions depicted by the black and green circles, respectively, in panel (b).The spectra have been normalised to the maximum intensity of the 2D-band.

Figure S6 .
Figure S6.(a,b) AFM characterisation of graphene before illumination (a) and after illumination (b) with Gaussian ( = ) beam (excitation power: 500 W).The root-mean-square roughness (R RMS ) of both panels (a) and (b) is displayed at the bottom left corner.(c,d) Variation in roughness (black) and waviness (green) across the white cut lines in (a) and (b), respectively.The rootmean-square roughness (R RMS ) and waviness (W RMS ) have been added at the bottom left corner.See Methods for details.

Figure S7 .
Figure S7.(a) Simulated intensity and phase front of LG beams in air for varying topological charges.(b) Phase fronts for  =  and  = - beams.

Figure
Figure S8.(a) Angular momentum (AM) flux in air as a function of topological charge for a transverse-normalised Laguerre-Gaussian beam.Excitation power: 50.(b) AM flux in air as a function of excitation power.

Figure S9 .
Figure S9.Simulated electric field intensity and temperature at graphene-substrate interface ( = .,power: ) for Gaussian beam and Laguerre-Gaussian beams of varying topological charges in monolayer graphene on SiO 2 /Si.A cut line (white line) of the electric field intensity and temperature rise with respect to the initial temperature   = . is displayed on the right-hand side.

Figure S10 .
Figure S10.Schematic of the PL set up which includes a 532nm laser and the resulting red photoluminescence from monolayer WS 2 .The LG beam is generated by inserting a spiral phase plate ( = , 532 nm, Vortex photonics) along the optical path.The beam is directed towards the sample with a 50:50 beam splitter and a 60X objective (NA=0.9).

Figure S11 .
Figure S11.(a) Reference spectra with no OAM (red) and right before OAM is introduced (black), showing no change between the two consecutive spectra.(b) Spectrum collected with OAM  =  (red) compared to the spectrum before OAM (black).(c,d) Spectrum collected with a Gaussian beam (c) 10 minutes and (d) 2 hours after OAM (red), compared to the spectrum before OAM (black).(e,f,g,h) Gaussian fits for the red spectra shown in panels (a,b,c,d) respectively.The green and red horizontal lines indicate changes in neutral exciton (X 0 ) and trion (T) emissions, respectively, and the T/X 0 intensity ratio is indicated at the top right corner.(i) Changes in the energies of the neutral exciton (X 0 ) obtained from the fittings shown in panels (e,f,g,h).The red shift in PL observed with LG beams across different positions in the CVD-grown monolayer and exfoliated monolayer flakes is shown in FigureS12aand FigureS12b, respectively.

Figure
Figure S12.(a) Shift with OAM for the neutral exciton (meV) as a function of T/X 0 intensity ratio in the CVD-grown monolayer.(b) Shift with OAM for the neutral exciton (meV) as a function of the initial position of the neutral exciton (meV) in exfoliated monolayer sample.The error bars were obtained from Gaussian fittings of experimental data.

Figure S13 .
Figure S13.Unnormalized PL spectra of an exfoliated monolayer WS 2 flake before and after OAM illumination.

Figure S14 .
Figure S14.OAM-induced red shift in PL collected from an exfoliated monolayer flake with the opposite sign of OAM ( = -) to that shown in the main text (see Figure 4).(a) Reference spectra with no OAM (red) and right before OAM is introduced (black), showing no change between the two consecutive spectra.(b) Spectrum collected with OAM  = - (red) compared to the spectrum before OAM (black).(c) Spectrum collected with a Gaussian beam after OAM (red), compared to the spectrum before OAM (black).

Figure
Figure S15.(a) Normalised PL spectrum of CVD-grown monolayer WS 2 on Al 2 O 3 as a function of temperature.(b) Energy positions for neutral exciton energy (black) and trion energy (red) as a function of temperature.(c) Trion-to-neutral exciton ratio as a function of temperature.The energy positions and T/X 0 ratio were obtained from Gaussian fittings of the experimental data in (a).

Figure
Figure S16.(a) Normalised PL spectrum of an exfoliated monolayer WS 2 flake on SiO 2 /Si as a function of temperature.(b) Energy positions for neutral exciton energy (black) and trion energy (red) as a function of temperature.(c) Trion-to-neutral exciton ratio as a function of temperature.The energy position and T/X 0 ratio were obtained from Gaussian fittings of the experimental data in (a).

Figure S17 .
Figure S17.Simulated electric field intensity and temperature for Gaussian and Laguerre-Gaussian beams of varying topological charges in monolayer WS 2 on Al 2 O 3 .Excitation power: .A cut line (white line) of the electric field intensity and temperature rise with respect to the initial temperature   = . is displayed on the right-hand side.

Figure S18 .
Figure S18.Simulated electric field intensity and temperature for Gaussian and Laguerre-Gaussian beams of varying topological charges in monolayer WS 2 on SiO 2 /Si.Excitation power: .A cut line (white line) of the electric field intensity and temperature rise with respect to the initial temperature   = . is displayed on the right-hand side.

( 18 )
Equation ( 18 ) is the continuity equation for angular momentum inside the medium: the divergence of the angular momentum flux density on the right-hand side is equated to the local rate of change of angular momentum on the left.This rate of change is given by the time derivative of the optical angular momentum density (both the torque due to the Lorentz force density  × ( • ∇) + ∂ ∂ × , and also the torque on each dipole about its own centre,  × ).